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arXiv:1002.3004 Twists of K-theory and TMF from arXiv Front: math.AT by Matthew Ando, Andrew J. Blumberg, David Gepner We explore an approach to twisted generalized cohomology from the point of view of stable homotopy theory and quasicategory theory provided by arXiv:0810.4535. We explain the relationship to the twisted K-theory provided by Fredholm bundles. We show how our approach allows us to twist elliptic cohomology by degree four classes, and more generally by maps to the four-stage Postnikov system BO0...4>. We also discuss Poincare duality and umkehr maps in this setting.
Stasheff sent and email to ALG-TOP on Aug 15, 2009, asking for info so that he could recover the history of twisted cohomology.
http://mathoverflow.net/questions/27567/twistings-for-other-cohomology-theories
nLab page on Twisted cohomology